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From: Virgil on 31 Oct 2006 16:24 In article <1c6ce$45470f63$82a1e228$20321(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Randy Poe wrote: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >>mueckenh(a)rz.fh-augsburg.de wrote: > >> > >>>stephen(a)nomail.com schrieb: > >>> > >>>>Can you describe a continuous version of the problem where each > >>>>"unit" of water has a well defined exit time? A key part of > >>>>the original problem is that the time at which each ball is > >>>>removed is defined and reached. This is crucial to the problem. It is > >>>>not just a matter of rates. If you added balls 1-10, then 2-20, > >>>>3-30, ... but you removed balls 2,4,6,8, ... then the vase is > >>>>not empty at noon, even though the rates of insertions and removals > >>>>are the same as in the original problem. So you cannot just > >>>>say the rate is 10 in and 1 out and base an answer on that. > >>> > >>>The answer for any time t *before noon* is independent of the chosen > >>>enumeration of the balls. Doesn't that fact make you think a bit > >>>deeper? > >> > >>And add this to the fact that noon and beyond cannot exist > >>in this problem. > > > > I'll ask you the same question I've been asking Tony about > > this "existence". > > > > Did noon exist yesterday? > > > > Is there anything that prevents me from defining a set > > of times t_n, n=1, 2, ... where t_n = 1/n minutes before > > noon yesterday? > > > > As soon as I define those times, does noon yesterday > > cease to exist? > > You may argue as much as you want. But the facts are quite simple. > > Talking about noon doesn't mean that you "have" noon. If you introduce > the notion of "time" in a problem, then that "time" should behave like > real (physical) time. And it is a fact that time cannot "flow" through > singularities. Does HdB have any experimental evidence to back up this claim? A description of such an experiment would be of great interest. But until some sort of physical evidence for HdB's physical claim is provided, it is quite in order to reject it as unfounded.
From: mueckenh on 31 Oct 2006 16:30 William Hughes schrieb: > > All entries of the list have a finite number of letters. > > Correct. And given any integer, we can take a set of lines such that > the number of letters in this set is greater than the integer. Correct. And given any set of lines we can find a finite number which is larger. Otherwise, at least one of the lines would contain a number of letters which was larger than any natural number, i.e. which was omega. > > > An infinite > > sequence is larger than any finite sequence. The diagonal of a list > > cannot have more letters than the lines. > > > > Correct. The number of letters in the lines is greater than any > integer. Incorrect. Greater than any integer is no integer but only omega (an its successors). > So the greatest number of letters the diagonal can have is > greater than any integer. Incorrect. See above. > > So the diagonal has infinite length (call this potentially infinite > length if > you get your kicks by saying potentially). The diagonal may have potentially infinite length, but that is less than actual infinity, i.e., omega which is the first number larger than any natural number. Die Anzahl einer unendlichen Menge [ist] eine durch das Gesetz der Zählung mitbestimmte unendliche ganze Zahl. (G. Cantor, Collected Works p. 174) .... kann also omega sowohl als eine gerade, wie als eine ungerade Zahl aufgefaÃ?t werden. (G. Cantor, Collected Works p. 178) Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu denken, welcher die Zahlen nu zustreben, wenn darunter nichts anderes verstanden wird, als daÃ?ï? omega die erste ganze Zahl sein soll, welche auf alle Zahlen folgt, d. h. gröÃ?er zu nennen ist als jede der Zahlen n. (G. Cantor, Collected Works p. 195) Regards, WM
From: mueckenh on 31 Oct 2006 16:34 David Marcus schrieb: > > Here it means, the result of my proof is valid for all possible > > theories which allow to define the infinite binary tree. > > Sorry. I still don't know what you mean. ZFC is enough to define an > infinite binary tree. Are you saying that your proof can be done in ZFC > or are you not saying this? If not, then what are you saying? If ZFC is enough to define an infinite binary tree, then it should be enough to see that there are more edges than paths. Where is a gap? is ZFC not able to sum the geometric series? Or is i not possible to consider fractions of elements? > > > Either you have a proof that can be given in ZFC or you don't. Which is > > > it? This shouldn't be a difficult question to answer. > > > > But it is not an interesting question. > > Perhaps not interesting to you, but please answer it anyway. I know of a really good logician who told me that he is unable to translate my proof into ZFC. I don't know what is missing. Neither does he. But, as I said, I will not learn to calculate horoscopes in order to judge that astrology is nonsense. Regards, WM > OK, let's use our logic (or standard logic or ZFC). The list has > infinitely many lines and columns, so the number of diagonal elements is > infinite. Your statement that the number of diagonal elements is finite > is wrong. You've jumped from "each entry is finite" to "the number of > columns is finite". Please give one letter which requires an actually infinite number omega of columns. If you can't, please stop the nonsense talk about an infinite number of finite numbers. > > If the list consists of finite sequences, then the diaogonal is a > > finite sequence too. Because it cannot be broader than the list > In ZFC or in some other system? Everywhere. Regards, WM
From: Virgil on 31 Oct 2006 16:35 In article <1162298787.575323.6800(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > The set of natural numbers is an infinite set that contains only finite > > numbers. > > Please do not assert over and over again this unsubstantiated nonsense Please do not reject over and over again axioms which you cannot falsify. > (this word means exactly what you think) but give an example, please, > of a natural number which does not belong to a finite sequence. Every member of every infinite sequence is also a member of infinitely many finite sequences, so what WM is saying is stupid. > If you > cannot do so, then it is obviously unnecessary to consider N as an > infinite sequence, because all its members belong to finite sequences. Stupidity compounded. Every finite sequence has a last member. That is what makes it finite. When one is dealing with a sequence which is prohibited from having have a last member, a finite sequence won't work.
From: mueckenh on 31 Oct 2006 16:36
Virgil schrieb: > In article <c5a1b$4545ba52$82a1e228$12545(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > Are you still doing physics, water, and a finite number of molecules? > > > Let us know when you switch to mathematics. > > > > I'm DOING mathematics. Mathematics is NOT independent of Physics. > > That dependency is a cross only physicists bear. > For non-physicists, mathematics is quite independent of physics. No. They only don't recognize the dependence. Regards, WM >> Every member of a list of only finite sequences is a finite sequence. > But that does not limit the number of such sequences to being finite. You say so. But then the number of such sequences is larger than the columns. However, it does not play any role. The diagonal has not more elements than the sequences. >> Every diagonal of such a list is a finite sequence. > Wrong! > Define the infinite sequence of finite strings in which the nth string > consists of n 1's, and define the infinite diagonal so that in its nth > position it has a "2" for each of infinitely many n's. It is ridiculous how the brain is washed by study of "logic". Did you ever hear of a finite set which contains an infinite set? Regards, WM |